The total number of ways to choose 4 marbles from 16 is: - GetMeFoodie
The Total Number of Ways to Choose 4 Marbles from 16: The Science Behind Combinations
The Total Number of Ways to Choose 4 Marbles from 16: The Science Behind Combinations
When faced with the question — “How many ways can you choose 4 marbles from a set of 16?” — many might initially think in terms of simple counting, but the true elegance lies in combinatorics. This article explores the exact mathematical answer, explains the concept of combinations, and reveals how you calculate the total number of ways to select 4 marbles from 16.
Understanding the Context
Understanding the Problem
At first glance, choosing 4 marbles from 16 might seem like a straightforward arithmetic problem. However, the key distinction lies in whether the order of selection matters:
- If order matters, you’re dealing with permutations — calculating how many ways marbles can be arranged when position is important.
- But if order doesn’t matter, and you only care about which marbles are selected (not the sequence), you’re looking at combinations.
Since selecting marbles for a collection typically concerns selection without regard to order, we focus on combinations — specifically, the number of combinations of 16 marbles taken 4 at a time, denoted mathematically as:
Image Gallery
Key Insights
$$
\binom{16}{4}
$$
What is a Combination?
A combination is a way of selecting items from a larger set where the order of selection is irrelevant. The formula to compute combinations is:
$$
\binom{n}{r} = \frac{n!}{r!(n - r)!}
$$
🔗 Related Articles You Might Like:
📰 Why Every True Anime Fan Must Know This Hi-Anime Masterpiece Now 📰 Unlock The Hidden World Of Hi-Anime You Never Knew Existed 📰 The One Hi-Anime That Adds Unbelievable Depth to Every Frame You Watch 📰 Lufthansa Stock 📰 How Many Oz Of Water Should I Drink 1738503 📰 New Details Enchantment Oblivion And The News Spreads 📰 Matlab On Ipad Now Compatible Easier To Use Than Everheres How 8099904 📰 Bank Of America Free Admission 📰 Home Epot 71064 📰 Artifact In Kythera 📰 Bankofamericq 📰 You Wont Believe The Hidden Gems In Girls Frontline Anime Hidden Gems You Need To See Now 934754 📰 Big Discovery 1Password 6 And It Stuns Experts 📰 Usg Corporation Stock 📰 Affordable Wireless Plans 📰 Lowest Interest Rate For Car Loan 2311136 📰 You Wont Believe How Windows 10 Assistant Saves You 10 Hours A Week 5004030 📰 Study Finds Virtual Computer Keyboard Mac And Authorities Take ActionFinal Thoughts
Where:
- $ n $ = total number of items (here, 16 marbles)
- $ r $ = number of items to choose (here, 4 marbles)
- $ ! $ denotes factorial — the product of all positive integers up to that number (e.g., $ 5! = 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1 = 120 $)
Using this formula:
$$
\binom{16}{4} = \frac{16!}{4!(16 - 4)!} = \frac{16!}{4! \cdot 12!}
$$
Note that $ 16! = 16 \ imes 15 \ imes 14 \ imes 13 \ imes 12! $, so the $ 12! $ cancels out:
$$
\binom{16}{4} = \frac{16 \ imes 15 \ imes 14 \ imes 13}{4 \ imes 3 \ imes 2 \ imes 1}
$$
Calculating the Value
Now compute the numerator and denominator:
-
Numerator:
$ 16 \ imes 15 = 240 $
$ 240 \ imes 14 = 3,360 $
$ 3,360 \ imes 13 = 43,680 $ -
Denominator:
$ 4 \ imes 3 = 12 $, $ 12 \ imes 2 = 24 $, $ 24 \ imes 1 = 24 $
